US6148056A - Efficient cone-beam reconstruction system using circle-and-line orbit data - Google Patents

Efficient cone-beam reconstruction system using circle-and-line orbit data Download PDF

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US6148056A
US6148056A US09/253,706 US25370699A US6148056A US 6148056 A US6148056 A US 6148056A US 25370699 A US25370699 A US 25370699A US 6148056 A US6148056 A US 6148056A
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Wen-Tai Lin
Mehmet Yavuz
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General Electric Co
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Priority to TR200000471A priority patent/TR200000471A2/xx
Priority to SG200000909A priority patent/SG82067A1/en
Priority to PL338536A priority patent/PL196564B1/pl
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T11/00—2D [Two Dimensional] image generation
    • G06T11/003—Reconstruction from projections, e.g. tomography
    • G06T11/006—Inverse problem, transformation from projection-space into object-space, e.g. transform methods, back-projection, algebraic methods
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B6/00—Apparatus or devices for radiation diagnosis; Apparatus or devices for radiation diagnosis combined with radiation therapy equipment
    • A61B6/52—Devices using data or image processing specially adapted for radiation diagnosis
    • A61B6/5258—Devices using data or image processing specially adapted for radiation diagnosis involving detection or reduction of artifacts or noise
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B6/00—Apparatus or devices for radiation diagnosis; Apparatus or devices for radiation diagnosis combined with radiation therapy equipment
    • A61B6/02—Arrangements for diagnosis sequentially in different planes; Stereoscopic radiation diagnosis
    • A61B6/027—Arrangements for diagnosis sequentially in different planes; Stereoscopic radiation diagnosis characterised by the use of a particular data acquisition trajectory, e.g. helical or spiral
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
    • Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y10S378/00—X-ray or gamma ray systems or devices
    • Y10S378/901—Computer tomography program or processor

Definitions

  • the field of the invention relates to computer tomography and more particularly to x-ray medical computer tomography.
  • Computer tomography (CT) using cone-beam projection technology typically employs an x-ray source to form a vertex in the shape of a cone.
  • a scanning orbit of the x-ray source is a curve along which the vertex of the cone beam moves during a scan.
  • a set of detectors are disposed at a fixed distance from the cone-beam source.
  • Methods currently exist which have established a relationship between cone-beam projections and a first derivative of the three dimensional (3D) Radon transforms of such projections.
  • each source position "S" can deliver Radon data positioned on a sphere, the Radon shell of "S.”
  • the corresponding spheres on a circular orbit sweep out the Radon space of the object, referred to as a torus. Regions inside the smallest sphere containing the object, but exterior to the torus, is called the shadow zone.
  • the shadow zone characterize the region where Radon data is missing from the circular orbit.
  • the line-orbit data is typically used for filling in the shadow zone. Even though the shadow zone is very small, as compared to the volume of the entire valid Radon data delivered by the circular orbit, current cone-beam projection techniques are computationally intensive. Accordingly, a need exists for simplifying image reconstruction based on the circle-and-line orbit.
  • a system for reconstructing x-ray tomographic images based upon the circle-and-line orbit.
  • the system includes the steps of combining, in the frequency domain, images reconstructed separately from the circular and linear orbit data.
  • the method involves the steps of (1) converting the shadow zone into a shadow cone in the frequency domain, (2) mapping the shadow cone in 3D Fourier space onto a set of 2D Fourier planes, (3) removing data lying within the region marked as the projection of the shadow cone from the Fourier transform of the circular orbit data and patching the same from the Fourier transform of the line orbit data, (4) transforming each 2D Fourier plane into a respective 2D image plane, and (5) converting the horn-shaped volume back to a grid volume.
  • An interpolation technique is also provided for reconstructing the line orbit data using a Direct Fourier Method (DFM).
  • DFM Direct Fourier Method
  • FIG. 1 depicts an x-ray tomography imaging system of the present invention
  • FIG. 2 depicts a cross-section of the circular orbit formed by the system of FIG. 1;
  • FIG. 3 depicts a shadow zone of FIG. 2 and corresponding shadow cone
  • FIGS. 4a and 4b illustrate a cross-section of the shadow cone of FIG. 2 projected onto a plane
  • FIG. 5 depicts incremental processing steps performed in the z-direction by the system of FIG. 1;
  • FIG. 6 depicts a set of parallel projection data at the ⁇ max angle (obtained from the M th row of the detector array as depicted in FIG. 5);
  • FIG. 7 depicts a set of parallel projection data at the ⁇ min angle (obtained from the first row of the detector array as depicted in FIG. 5).
  • FIG. 8 depicts inverse bilinear interpolation of the system of FIG. 1.
  • FIG. I depicts a CT imaging system 10, generally, in accordance with an illustrated embodiment of the invention. Included within the system 10 is a central processing unit (CPU) 16, a sampling device 12, and other image support and processing subsystems 14, 18, 20, 22.
  • the sampling device 12 includes an x-ray source 34 and an image detecting array 32 along with actuating devices.
  • the image support devices 14, 18, 22 include a Fast Fourier processor 14, an interpolator 18, a memory 20, and a display 22.
  • Direct Fourier Method back projection is based on the projection slice theorem, which postulates that a one dimensional (1D) Fourier Transform (FT) of a projection at a specific angle corresponds to the cross section of the 2D FT of the object at the same angle.
  • DFM shall be used as an alternative to the conventional back-projection reconstruction involving the line orbit.
  • the first two elements, f c0 (r) and f c1 (r), are evaluated using circularly-scanned data and techniques (e.g., as given by Feldkamp et al. and Hu).
  • Feldkamp et al. has published cone-beam algorithms in a paper entitled, PRACTICAL CONE-BEAM ALGORITHM, published in the Journal of the Optical Society of America, Vol. 1, No. 6, June 1984, pages 612-619, herein incorporated by reference.
  • the second element is needed only when the third term is evaluated in the Radon space. Since we will evaluate the third term in the Fourier space, thus the second term will be ignored.
  • the third element, f 1 (r), is evaluated from the linear orbit data. It is typically used for filling in a shadow zone 28 defined by a torus 24, which is swept out by the circular orbit about the axis of rotation, and the smallest sphere containing the object 40 (FIG. 2). Even though this shadow zone 28, which is defined in Radon space, is small as compared to the torus 24, the current back-projection process is computationally intensive, as is apparent from the complexity of f 1 (r) as defined, for example in a paper by Hui Hu, entitled, CONE BEAM RECONSTRUCTION ALGORITHM FOR THE CIRCULAR ORBIT AND A CIRCLE-AND-LINE ORBIT, published by the Applied Science Laboratory, GE Medical Systems, Jan.
  • the present invention describes a method for determining a computational efficient CT reconstruction using data obtained from a line orbit scan of the object 40 (illustrated in FIG. 2).
  • the shadow zone 28 in Radon space is augmented and converted into a shadow cone 30 in frequency space (FIG. 3).
  • the patching process performed by CPU 16 is simplified from the original three-dimensional frequency space into multiple 2D frequency slices.
  • the first step is based on the fact that the Fourier transform of a radial line in 3D Radon space is equivalent to the same radial line in 3D Fourier space. If a radial line in 3D Radon space intercepts the shadow zone 28, then its counterpart in the 3D Fourier space is "contaminated" because some Radon data in the radial Radon line has been missing. Since all the radial lines intercepting the shadow zone 28 form a shadow cone 30, the collection of their counterparts in the 3D Fourier space also form a frequency-insufficient cone. The frequency-insufficient cone (or the contaminated cone) of the first term of equation (1), which is defined in Fourier space, will be replaced by that of the third term of equation (1).
  • the second step is to show that the linear orbit supports enough frequency data to replace the contaminated cone (which is due to the circular orbit), followed by a procedure for replacing the contaminated cone in 2D Fourier space.
  • FIG. 5 shows that the linear scanning orbit may be implemented by holding an x-ray source 34 and a detector panel 32 stationary, while moving object 40 along the z-axis 36 (which is also the rotational axis of the circular orbit).
  • the object 40 is shown to move in the "z direction" in discrete increments identified at "z -- step” 36 increments.
  • Each vertical column of detectors 32 together with the source 34 defines a plane through the object 40.
  • every detector position defines a specific angle and the data collected by that detector corresponds to a parallel projection of the object sampled by intervals of z-step 36.
  • the shadow cone 30 (FIG. 3) in 3D Fourier space may be projected as a 2D shadow cone onto a set of 2D Fourier planes.
  • the inverse transform of each of these 2D Fourier planes corresponds to an image plane defined by the x-ray source and a column "N" of detectors 32 (FIG. 5).
  • F(u,v) be the Fourier transform of a 2D function f(x,y).
  • F(u',v',w) is obtained by rotating the (u,v) coordinates by an angle ⁇ , as illustrated in FIG. 4a, while keeping the w-coordinate unchanged.
  • the rotated coordinate system is graphically illustrated in FIGS. 4a and 4b, where the new coordinates, (u',v',w), are chosen such that u' is parallel to the normal of the plane of interest (shown as the SO' line).
  • the shadow disc 26 which is a cross-section of shadow cone 30 (FIG. 4a), intersecting a horizontal plane 27 (FIG. 4b), is projected onto SO' as a line segment centered at "O'" and having end points "f" and "e.”
  • the length of the line segment fe equals the diameter of the shadow cone 30 centered at "O.”
  • shadow cone 30 (FIG. 4a), defining the area in Radon space in 3D Fourier space, is projected onto 2D Fourier subspace by applying a 1D inverse Fourier transform.
  • the scanned object 40 is represented in a series of 2D Fourier planes, and replaces the projected 2D shadow disc 26 in all the individual planes.
  • Detector array 32 (FIG. 5) will be referred to by the expression, D(M ⁇ N), where the detector array 32 has "M" rows and "N” columns.
  • Projection data of circular orbit will be referred to by the expression P C (M ⁇ N ⁇ V C ), where V C is the total number of views.
  • Projection data of the line orbit will be referred to by the expression P I (M ⁇ N ⁇ V I ), where V I is also the total number of views.
  • An image reconstructed from circular orbit data will have the form, I C (K ⁇ K ⁇ K) or I CS (K ⁇ N ⁇ K); the former is a cubic volume, while the latter is a "horn-shaped" volume defined by the x-ray source and the detector panel.
  • An image reconstructed from line orbit data will have the form, I l (K ⁇ K ⁇ K) or I lS (K ⁇ N ⁇ K); likewise, the former is a cubic volume and the latter is a "horn-shaped" volume.
  • the image is reconstructed from the line-orbit data.
  • the line-orbit projection data is modified to accommodate the effect of divergence as a plurality of rays passes from source 34 to detector array 32, as graphically illustrated in FIGS. 5, 6, and 7.
  • the divergent ray detected on the detector panel is modified by multiplying the detector values with a weighting function as follows: ##EQU6## where "D” in this case is the source-to-detector distance (FIG. 5), and wherein ISO is projected onto (m 0 ,n 0 ) of the detector panel 32.
  • the values "a” and "b” are the detector pitches in the row and column dimensions, respectively. Detector pitches are standard definitions defining the distance between the centers of two adjacent detectors.
  • the n th angular vertical plane is reconstructed next.
  • the projection data, P ln (M ⁇ V 1 ) is extracted from the n th column of the P l array, where the m th row corresponds to a fixed beam angle of ⁇ m .
  • either one of two methods may be taken.
  • An inverse FFT is utilized to convert the Fourier data set to object space.
  • the final image may be reconstructed by one of two possible methods.
  • the shadow cone 30 may be replaced in 3D Fourier space.
  • the shadow cone 30 may be replaced in 2D Fourier spaces and then interpolated from the horn shape volume back to cubic volume.
  • the replacement of the shadow cone 30 in 3D space is presented first.
  • the Feldkamp algorithm is used to reconstruct the image, I C , associated with torus 24, from the circular orbit data.
  • a 3D Fourier transform, F(I C ) is obtained from I C .
  • the horn shaped volume, I I .sbsb.S is interpolated to obtain a corresponding cubic volume, I I .
  • a 3D Fourier transform, F(I I ) is obtained from I I .
  • the shadow cone portion of F(I C ) is replaced with its counterpart from F(I I ) to provide a repaired Fourier data array.
  • An inverse Fourier transform is then applied to the repaired Fourier data array to return to cubic space.
  • the Fourier transform F(I CS ) and F(I I .sbsb.S) is obtained from I CS and I I .sbsb.S.
  • the shadow cone is substituted into each 2D plane of F(I CS ), with its counterpart defined in F(I I .sbsb.S).
  • An inverse Fourier transform is applied to each of the repaired 2D sets of Fourier data.
  • the horn-shape volume is interpolated back to cubic volume.
  • the DFM method described above requires interpolation from polar coordinates to rectangular coordinates.
  • the cone beam reconstruction described above is based on circle and line orbits which requires that the missing shadow zone 28 of circle orbit data in the Fourier domain to be replaced by the line orbit data.
  • the object 40 is shown to move in the "z direction" in discrete increments identified as "z -- step” 36 increments.
  • Each vertical column of detectors 32 together with the source 34 defines a plane through the object 40. Within each vertical plane, every detector position defines a specific angle and the data collected by that detector corresponds to a parallel projection of the object sampled by intervals of z-step (FIGS. 5, 6, and 7).
  • One approach is to reconstruct the filtered back-projection (FBP) image for each angular plane using the parallel projections from the set of limited angular views and then to compute the 2D FT and use it to replace the missing angular segment.
  • FBP filtered back-projection
  • Another alternative is to convert the polar grid data from 1D FT of parallel projections to rectangular grid directly in 2D FT domain. This method is similar to the interpolation required in Direct Fourier Method back-projection.
  • ⁇ .sub. ⁇ (x, y) be the image in a fixed coordinate system x-y and ⁇ .sub. ⁇ (x, y) represent the object in a coordinate system x-y rotated from x-y by an angle ⁇ .
  • the projection of the object at a view angle ⁇ is
  • M.sub. ⁇ ( ⁇ , ⁇ ) denotes the 2D FT of ⁇ .sub. ⁇ (x, y) in polar coordinates
  • the object is reconstructed by 2D inverse FT using the expression,
  • M( ⁇ , ⁇ ) is not known at all positions but only at finite set of discrete points ( ⁇ j , ⁇ k ). The problem then becomes interpolating from the known values of polar points to the values requires over a rectangular Cartesian grid.
  • x(t) be a periodic function in time space "t" with period "T.” If x(t) is angularly band-limited to a value ##EQU9## where "c" is a constant, and "K” is an integer relating to the number of samples of x(t), then x(t) can be written as: ##EQU10##
  • the parallel projection data are sampled in intervals of a predetermined z-step 36.
  • a predetermined z-step window for example 2A
  • the "z -- step" 36 is selected to be small enough such that the effective bandwidth, ##EQU16## [points/cycle], will substantially include the frequency content of the projections.
  • Another source of error is inadequate sampling in a radial frequency. If the object is space limited in diameter to ##EQU17## its projections will also be space limited to 2A, as such, the FT is sampled uniformly at intervals of at least, ##EQU18## [points/cycle], in order to avoid aliasing.
  • the effective bandwidth is defined as "B.”
  • N the number of radial samples "N” are chosen such that ##EQU19## Also the grid spacing in the 2D FT domain is at most ##EQU20## [points/cycle].
  • Non-isotropic and off-centered objects will not be angularly band-limited. So, the number of projections may be very large. However, in practice the number of projections is chosen such that in the 2D FT domain the sampling between two neighboring radial lines will be approximately equal to the maximum grid spacing of ##EQU21## [points/cycle].
  • an inverse bilinear interpolation technique is used on data in the polar grid format.
  • ⁇ ( ⁇ ) the function defined in equation (13) may be used.
  • this function is computationally impractical.
  • values in the interpolation function must be truncated which causes artifacts.
  • NN Nearest neighbor
  • Convolution with a rectangle function in the spatial domain is equivalent to multiplying the signal in frequency domain by a sinc(. ) function, which is a poor low pass filter since it has prominent side lobes.
  • Linear interpolation amounts to convolution of the sampled data by a triangle window.
  • This function corresponds in frequency domain to a sinc 2 (. ) function which has a better low pass filtering property than NN.
  • the NN method interpolates on the basis of a single point.
  • the linear interpolation algorithm interpolates on the basis of two nearest points. Using three points for interpolation would result in two points on one side and only one on the other side.
  • the B-spline method enables interpolation over the two nearest points in each direction.
  • the Cubic B-spline method comprises four convolutions of the rectangular function of the NN method. Since Cubic B-splines are symmetric, they only need to be defined in the interval (0, 2). Mathematically, the B-splines can be written as: ##EQU23##
  • the Cubic B-splines equations defined above may be adapted to fit a smooth third-order polynomial through the available set of shadow cone 30 data.
  • Another interpolation function involves the truncation of ⁇ ( ⁇ ).
  • ⁇ ⁇
  • the ⁇ ( ⁇ ) function has considerable energy and, as such, does not decay fast over an extended distance, and therefore cannot be easily implemented as a space domain convolution. It is natural to truncate the function over a small distance, but truncation disregards much of the above said energy.
  • truncation in the space domain will produce ringing artifacts when the object is converted from the space domain to the signal domain.
  • Bilinear interpolation is a 2D interpolation method.
  • f(x,y) is evaluated by a linear combination of f(n 1 ,n 2 ) at the four closest points for n 1 T ⁇ x ⁇ (n 1 +1)T and n 2 T ⁇ y ⁇ (n 2 +1)T.
  • the interpolated value f(x,y) in the bilinear method is: ##EQU24##
  • Inverse bilinear interpolation converts polar-grid data into rectangular-grid data via inverse bilinear weighting. That is, each polar-grid element is redistributed to its four nearest rectangular grids using the four sub-areas as normalized weights.
  • the method is robust in the sense that the original data does not have to be regularly distributed, as long as the data has at least one element in each rectangular square. As such, undesirable data elements caused by defects or seams in detector array 32 (FIG. 5) may be ignored.
  • Another benefit of this process is that all the good data are fully utilized, which, consequently, provides a better signal-to-noise ratio (SNR).
  • SNR signal-to-noise ratio
  • a linear orbit datum value "x" and associated grid points used in inverse bilinear interpolation are graphically illustrated in FIG. 8.
  • the data value "x" is distributed to its four nearest grid points as follows:
  • Each grid point has an accumulated value, g ij , which identifies the accumulated weighted "x" values from the four adjacent squares.
  • g ij identifies the accumulated weighted "x" values from the four adjacent squares.
  • w ij which identified the accumulated weights (a ij ) associated with the bilinear distribution of pixel (I,j). In fact, this is a normalization step at the end of the interpolation procedure because each grid value is divided by accumulated weight, w ij .
  • Truncated ⁇ ( ⁇ ) interpolation method was also used in the azimuthal direction (including 8 neighboring points). Linear interpolation was used in the radial direction. Nearest neighbor interpolation and Linear interpolation generated better performance results than Cubic B-spline interpolation. However, Cubic B-spline uses only 4 points as opposed to 8 points in Truncated ⁇ ( ⁇ ) interpolation. So, Cubic B-spline is computationally more attractive than the 8 points Truncated ⁇ ( ⁇ ) interpolation, as such, Truncated ⁇ ( ⁇ ) interpolation was not included in the comparison.
  • Polar points based bilinear interpolation was also evaluated.
  • the 2D FT of the slice of interest was computed and used as the base reference for of comparing the above identified methods.
  • the 2D FT of the FBP reconstruction was also included in the comparison.
  • FBP reconstruction requires the reconstruction of the 2D FMP image and then computation of the 2D FT.
  • Polar bilinear interpolation does not require conversion back to the image domain. This results in a computational advantage over the FBP method.
  • the inverse bilinear interpolation has less error both in the 2D FT domain and in the image domain compared to the other listed interpolation methods.
  • the number of views in the 2D FT domain and the image domain may also be reduced from 256 without significantly increasing the image error as, for example, can be observed by the data in Table VI.

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US09/253,706 US6148056A (en) 1999-02-22 1999-02-22 Efficient cone-beam reconstruction system using circle-and-line orbit data
IL13454200A IL134542A (en) 1999-02-22 2000-02-15 Efficient cone-beam reconstruction system using circle-and-line orbit data
DE2000630498 DE60030498T2 (de) 1999-02-22 2000-02-16 Effizientes Kegelstrahl-Rekonstruktionssystem mittels Daten von kreis- und linienförmigen Quellentrajektorien.
EP20000301212 EP1031943B1 (en) 1999-02-22 2000-02-16 Efficient cone-beam reconstruction system using circle-and-line orbit data
TR200000471A TR200000471A2 (tr) 1999-02-22 2000-02-21 Daire-ve-doğru yörünge verileri kullanılarak verimli konik-ışın oluşturulması.
SG200000909A SG82067A1 (en) 1999-02-22 2000-02-21 Efficient cone-beam reconstruction system using circle-and-line orbit data
PL338536A PL196564B1 (pl) 1999-02-22 2000-02-21 Sposób i urządzenie do rekonstruowania rentgenowskich obrazów tomograficznych
JP2000042496A JP4519974B2 (ja) 1999-02-22 2000-02-21 Ｘ線断層画像を再構成する方法及び装置
BR0000899A BR0000899B1 (pt) 1999-02-22 2000-02-22 mÉtodo e aparelho de reconstruir imagens tomogrÁficas de raios x.

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US6546067B2 (en) * 2001-01-30 2003-04-08 Kabushiki Kaisha Toshiba Reconstruction and scan of 4D-CT
US6560308B1 (en) * 2001-10-26 2003-05-06 Kabushiki Kaisha Toshiba Method and system for approximating missing data in cone beam x-ray CT reconstruction
KR100397553B1 (ko) * 2002-06-20 2003-09-13 (주)메비시스 경사진 연속 단면영상의 볼륨데이터 생성방법
US6674913B1 (en) * 2000-03-23 2004-01-06 Kwangju Institute Of Science And Technology Method and apparatus for obtaining a high resolution image
US20050152494A1 (en) * 2001-08-16 2005-07-14 Alexander Katsevich Efficient image reconstruction algorithm for the circle and arc cone beam computer tomography
US20050175142A1 (en) * 2004-02-09 2005-08-11 Xiangyang Tang Method and apparatus for obtaining data for reconstructing images of an object
US20060029180A1 (en) * 2001-08-16 2006-02-09 Alexander Katsevich Exact filtered back projection (FBP) algorithm for spiral computer tomography with variable pitch
US20070140408A1 (en) * 2005-12-05 2007-06-21 Yasuro Takiura X-ray ct imaging method and x-ray ct apparatus
US20070248255A1 (en) * 2006-04-25 2007-10-25 Guang-Hong Chen System and Method for Estimating Data Missing from CT Imaging Projections
US20090097612A1 (en) * 2007-10-12 2009-04-16 Siemens Medical Solutions Usa, Inc. System for 3-Dimensional Medical Image Data Acquisition
EP2059171A2 (en) * 2006-08-31 2009-05-20 Koninklijke Philips Electronics N.V. Imaging system
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